matematicas visuales visual math

If we start with a segment we can draw a regular pentagon, only using ruler and compass, that has this segment as one side.

We already know that the diagonal of a regular pentagon are in golden ratio to its sides and that the golden ratio is denoted by and its value is:

This is the basic step:

Drawing a regular pentagon with ruler and compass: firsts steps | matematicasVisuales

Then the value of a is:

Drawing a regular pentagon with ruler and compass: using Pythagorean Theorem | matematicasVisuales

Then we can draw a segment with length the diagonal of the regular pentagon:

Drawing a regular pentagon with ruler and compass: the diagonal and the golden ratio | matematicasVisuales
Drawing a regular pentagon with ruler and compass: the diagonal and the golden ratio | matematicasVisuales

And we can finish our work and get a regular pentagon:

Drawing a regular pentagon with ruler and compass: finishing the pentagon | matematicasVisuales

Drawing twelve pentagons we get the net of a dodecahedron:

Drawing a regular pentagon with ruler and compass: a plane net of a dodecahedron | matematicasVisuales

MORE LINKS

Pythagoras Theorem: Euclid's demonstration
Demonstration of Pythagoras Theorem inspired in Euclid.
Durer's approximation of a Regular Pentagon
In his book 'Underweysung der Messung' Durer draw a non-regular pentagon with ruler and a fixed compass. It is a simple construction and a very good approximation of a regular pentagon.
The golden rectangle
A golden rectangle is made of an square and another golden rectangle.
The golden rectangle and the dilative rotation
A golden rectangle is made of an square an another golden rectangle. These rectangles are related through an dilative rotation.
The golden spiral
The golden spiral is a good approximation of an equiangular spiral.
The golden rectangle and two equiangular spirals
Two equiangular spirals contains all vertices of golden rectangles.
The icosahedron and its volume
The twelve vertices of an icosahedron lie in three golden rectangles. Then we can calculate the volume of an icosahedron
Regular dodecahedron
Some properties of this platonic solid and how it is related to the golden ratio. Constructing dodecahedra using different techniques.
Durer and transformations
He studied transformations of images, for example, faces.
Plane developments of geometric bodies: Dodecahedron
The first drawing of a plane net of a regular dodecahedron was published by Dürer in his book 'Underweysung der Messung' ('Four Books of Measurement'), published in 1525 .
Dilation and rotation in an equiangular spiral
Two transformations of an equiangular spiral with the same general efect.
Equiangular spiral
In an equiangular spiral the angle between the position vector and the tangent is constant.
Leonardo da Vinci: Drawing of a dodecahedron made to Luca Pacioli's De divina proportione.
Leonardo da Vinci made several drawings of polyhedra for Luca Pacioli's book 'De divina proportione'. Here we can see an adaptation of the dodecahedron.
Dilative rotation
A Dilative Rotation is a combination of a rotation an a dilatation from the same point.